Expand the Trigonometric Expression cos((3pi)/4-x)

$\cos (\frac{3 π}{4} - x)$

Step 1

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$\cos (\frac{3 π}{4}) \cos (x) + \sin (\frac{3 π}{4}) \sin (x)$

Step 2

Simplify each term.

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- \cos (\frac{π}{4}) \cos (x) + \sin (\frac{3 π}{4}) \sin (x)$

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{2}}{2} \cos (x) + \sin (\frac{3 π}{4}) \sin (x)$

Combine $\cos (x)$ and $\frac{\sqrt{2}}{2}$ .

$- \frac{\cos (x) \sqrt{2}}{2} + \sin (\frac{3 π}{4}) \sin (x)$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- \frac{\cos (x) \sqrt{2}}{2} + \sin (\frac{π}{4}) \sin (x)$

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\cos (x) \sqrt{2}}{2} + \frac{\sqrt{2}}{2} \sin (x)$

Combine $\frac{\sqrt{2}}{2}$ and $\sin (x)$ .

$- \frac{\cos (x) \sqrt{2}}{2} + \frac{\sqrt{2} \sin (x)}{2}$

Expand the Trigonometric Expression cos((3pi)/4-x)

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